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AUTHOR: 



HINTON, CHARLES H. 



TITLE: 



A LANGUAGE OF SPACE 



PLACE: 



LONDON 



DA TE : 



1906 



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Hinton, Gh-rxrlo;; Hovmrd, 186S-1907. 

/V languaco of space, by C. H. Hinton... London, 
Sonnenr.gli.oin, IdOC, ' ' 

!4 p. diniiro. 18.^ en in 2&J- cm, 

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9 



iT LANGUAGE 
OF SPACE 



KY. 



BY C. H. HINTON 






The following pages contain an outline of a 
proposed language of space, and indicate the uses 
and applications of such a language. 

While written immediately as an addition to a 
book— "The Fourth Dimension," Second Edition, 
1906, Appendix IL— on a special province of space- 
thinking, they present the subject of a space language 
in a sufficiently clear manner to obviate the necessity 
of a separate presentation, and will 1 hope lead to the 
adoption of a recognised nomenclature. 



I 



LONDON 

^WAN SONNENSCHEIN & CO., LIMITED 
25 HIGH STREET, BLOOMSBURY 

1906 



ii 



A LANGUAGE OF SPACE 

The mere naming the parts of the figures we consider 
involves a certain amount of time and attention. This 
time and attention leads to no result, for with each 
new figure the nomenclature applied is completely 
changed, every letter or symbol is used in a diflferent 
significance. 

Surely it must be possible in some way to utilise the 
labour thus at present wasted ! 

Why should we not make a language for space itself, so 
that every position we want to refer to would have its own 
name ? Then every time we named a figure in order to 
demonstrate its properties we should be exercising 
ourselves in the vocabulary of place. 

If we use a definite system of names, and always refer 
to the same space position by the same name, we create 
as it were a multitude of little hands, each prepared to 
grasp a special point, position, or element, and hold it 
for us in its proper relations. 

We make, to use another analogy, a kind of mental 
paper, which has somewhat of the properties of a sensitive 
plate, in that it will register, without efibrt, complex, 
visual, or tactual impressions. 

But of far more importance than the applications of a 
space language to the plane and to solid space is the 



? 



, » 



* . 



' > 
o 



A LANGUAGE OK SPACE o 

facilitation it brings with it to the study of four-dimen- 
sional shapes. 

I have delayed introducing a space language because 
all the systems I made turned out, after giving them a 
fair trial, to be intolerable. I have now come upon one 
which seems to present features of permanence, and I will 
here give an outline of it, so that it can be applied to 
the subject of the text, and in order that it may be 
subjected to criticism. 

The principle on which the language is constructed is 
to sacrifice every other consideration for brevity. 

It is indeed curious that we are able to talk and 
converse on every subject of thought except the funda- 
mental one of space. The only way of speaking about 
the spatial configurations that underlie every subject 
of discursive thought is a co-ordinate system of numbers. 
This is so awkward and incommodious that it is never 
used. In thinking also, in realising shapes, we do not 
use it; we confine ourselves to a direct visualisation. 

Now, the use of words corresponds to the storing up 
of our experience in a definite brain structure. A child, 
in the endless tactual, visual, mental manipulations it 
makes for itself, is best left to itself, but in the course 
of instruction the introduction of space names would 
make the teachers work more cumulative, and the child's 
knowledge more social. 

Their full use can only be appreciated, if they are 
introduced early in the course of education ; but in a 
minor degree any one can convince himself of their 
utility, especially in our immediate subject of handling 
four-dimensional shapes. The sum total of the results 
obtained in the preceding pages can be compendiously and 
accurately expressed in nine words of the Space Language. 
In one of Plato's dialogues Socrates makes an experi- 
ment on a slave boy standing by. He makes certain 



to 



A LANGUAGE OF SPACE 



A LANGUAGE OF SPACE 



perceptions of space awake in the mind of Meno*s slave 
by directing his close attention on some simple facts of 
geometry. 

By means of a few words and some simple forms we can 
repeat Plato's experiment on new ground. 

Do we by directing our close attention on the facts of 
four dimensions awaken a latent faculty in ourselves? 
The old experiment of Plato's, it seems to me, has come 
down to us as novel as on the day he incepted it, and its 
significance not better understood through all the dis- 
cussion of which it has been the subject. 

Imagine a voiceless people living in a region where 
everything had a velvety surface, and who were thus 
deprived of all opportunity of experiencing what sound is. 
They could observe the slow pulsations of the air caused 
by their movements, and arguing from analogy, they 
would no doubt infer that more rapid vibrations were 
possible. From the theoretical side they could determine 
all about these more rapid vibrations. They merely differ, 
they would say, from slower ones, by the number that 
occur in a given time; there is a merely formal difference. 

But suppose they were to take the trouble, go to the 
pains of producing these more rapid vibrations, then a 
totally new sensation would fall on their rudimentary ears. 
Probably at first they would only be dimly conscious of 
Sound, but even from the first they would become aware 
that a merely formal difference, a mere difference in point 
of number in this particular respect, made a great difference 
practically, as related to them. And to us the difference 
between three and four dimensions is merely formal, 
numerical. We can tell formally all about four dimensions^ 
calculate the relations that would exist. But that the 
difference is merely formal does not prove that it is a 
futile and empty task, to present to ourselves as closely as 
we can the phenomena of four dimensions. In our formal 






\ 



X, 



V 



f« 



knowledge of it, the whole question of its actual relation 
to us, as we are, is left in abeyance. 

Possibly a new apprehension of nature may come to U3 
through the practical, as distinguished from the mathe- 
matical and formal, study of four dimensions. As a child 
handles and examines the objects with which he comes in 
contact, so we can mentally handle and examine four- 
dimensional objects. The point to be determined is this. 
Do we find something cognate and natural to our faculties, 
or are we merely building up an artificial presentation of 
a scheme only formally jx)ssible, conceivable, but which 
has no real connection with any existing or possible 



experience 



'? 



This, it seems to me, is a question which can only be 
settled by actually trying. This practical attempt is the 
logical and direct continuation of the experiment Plato 
devised in the "Meno/* 

Why do we think true? Why, by our processes of 
thought, can we predict what will happen, and correctly 
conjecture the constitution of the things around us ? 
This is a problem which every modern philosopher has 
considered, and of which Descartes, Leibnitz, Kant, to 
name a few, have given memorable solutions. Plato wa? 
the first to suggest it. And as he had the unique position 
of being the first devisor of the problem, so his solution 
is the most unique. Later philosophers have talked about 
consciousness and its laws, sensations, categories. But 
Plato never used such words. Consciousness apart from a 
conscious being meant nothing to him. His was always 
an objective search. He made man's intuitions the basif 
of a new kind of natural history. 

In a few simple words Plato puts us in an attitude 
with regard to psychic phenomena — the mind — the ego — 
"what we are," which is analogous to the attitude scientific 
men of the present day have with regard to the phenomena 



6 



A LANGUAGE OF SPACE 



of outward nature. Behind this first apprehension of ours 
of nature, there is an infinite depth to be learned and 
known. Plato said that behind the phenomena of mind 
that Meno's slave boy exhibited, there was a vast, an 
infinite perspective. And his singularity, his originahty, 
comes out most strongly marked in this, that the i)er- 
spective, the complex phenomena beyond were, according 
to him, phenomena of personal experience. A footprint 
in the sand means a man to a being that has the con- 
ception of a man. But to a creature that has no such 
conception, it means a curious mark, somehow resulting 
from the concatenation of ordinary occurrences. Such a 
being would attempt merely to explain how causes known 
to him could so coincide as to produce such a result ; 
he would not recognise its significance. 

Plato introduced the conception which made a new 
kind of natural history possible. He said that Meno's 
slave boy thought true about things he had never 
learned, because his " soul " had experience. I know this 
will sound absurd to some people, and it flies straight 
in the face of the maxim, that explanation consists in 
showing how an eflfect depends on simple causes. But 
what a mistaken maxim that is! Can any single instance 
be shown of a simple cause? Take the behaviour of 
spheres for instance ; say those ivory spheres, billiard balls, 
for example. We can explain their behaviour by supposing 
they are homogeneous elastic solids. We can give formulae 
which will account for their movements in every variety. 
But are they homogeneous elastic solids ? No, certainly 
not. They are complex in physical and molecular structure, 
and atoms and ions beyond open an endless vista. Our 
simple explanation is false, false as it can be. The balls 
act as if they were homogeneous elastic spheres. There is 
a statistical simplicity in the resultant of very complex 
conditions, which makes that artificial conception useful. 



1 



( 



»i 



, « 



A LAKGUAGE OF SPACE 7 

But its usefulness must not blind us to the fact that it is 
artificial. If we really look deep into nature, we find a 
much greater complexity than we at first suspect. And 
so behind this simple '*!," this myself, is there not a 
parallel complexity ? Plato's " soul " would be quite 
acceptable to a large class of thinkers, if by " soul " and 
the complexity he attributes to it, he meant the product 
of a long course of evolutionary changes, whereby simple 
forms of living matter endowed with rudimentary sensation 
had gradually developed into fully conscious beings. 

But Plato does not mean by •* soul " a being of such a 
kind. His soul is a being whose faculties are clogged by 
its bodily environment, or at least hampered by the 
difficulty of directing its bodily frame — a being which 
is essentially higher than the account it gives of itself 
through its organs. At the same time Plato's soul is 
not incorporeal. It is a real being with a real experience. 
The question of whether Plato had the conception of non- 
spatial existence has been much discussed. The verdict 
is, I believe, that even his " ideas " were conceived by him 
as beings in space, or, as we should say, real. Plato's 
attitude is that of Science, inasmuch as he thinks of a 
world in Space. But, granting this, it cannot be denied 
that there is a fundamental divergence between Plato's 
conception and the evolutionary theory, and also an 
absolute divergence between his conception and the 
genetic account of the origin of the human faculties. 
The functions and capacities of Plato's ** soul " are not 
derived by the interaction of the body and its environment. 

Plato was engaged on a variety of problems, and his 
religious and ethical thoughts were so keen and fertile 
that the experimental investigation of his soul appears 
involved with many other motives. In one passage Plato 
will combine matter of thought of all kinds and from all 
sources, overlapping, interrunning. And in no case is he 



8 



A LANGUAGE OF SPACE 



A LANGUAGE OF SPACE 



9 



more involved and rich than in this question of the soul. 
In fact, I wish there were two words, one denoting that 
being, corporeal and real, but with higher faculties than 
we manifest in our bodily actions, which is to be taken as 
the subject of experimental investigation ; and the other 
word denoting " soul '* in the sense in which it is made 
the recipient and the promise of so much that men desire^ 
It is the soul in the former sense that I wish to investigate^ 
and in a limited sphere only. I wish to find out, in con- 
tinuation of the experiment in the Meno, what the ** soul "" 
in us thinks about extension, experimenting on the 
grounds laid down by Plato. He made, to state the 
matter briefly, the hypothesis with regard to the thinking 
power of a being in us, a *' soul." This soul is not acces- 
sible to observation by sight or touch, but it can be 
observed by its functions ; it is the object of a new kind 
of natural history, the materials for constructing which 
lie in what it is natural to us to think. With Plato- 
*' thought " was a very wide-reachiog term, but still I 
would claim in his general plan of procedure a place for 
the particular question of extension. 

The problem comes to be, " What is it natural to us to- 
think about matter qua extended ? " 

First of all, I find that the ordinary intuition of any 
simple object is extremely imperfect. Take a block of 
differently marked cubes, for instance, and become ac- 
quainted with them in their positions. You may think 
you know them quite well, but when you turn them round 
— rotate the block round a diagonal, for instance — you 
will find that you have lost track of the individuals in 
their new positions. You can mentally construct the- 
block in its new position, by a rule, by taking the remem- 
bered sequences, but you don't know it intuitively. By 
observation of a block of cubes in various positions, and 
very expeditiously by a use of Space names applied to the 



*^ 



♦ t 



• / 



cubes in their different presentations, it is possible to get 
an intuitive knowledge of the block of cubes, which is not 
•disturbed by any displacement. Now, with regard to this 
intuition, we modems would say that I had formed it by 
my tactual visual experiences (aided by hereditary pre- 
disposition). Plato would say that the soul had been 
stimulated to recognise an instance of shape which it 
knew. Plato would consider the operation of learning 
merely as a stimulus; we as completely accounting for 
the result. The latter is the more common-sense view. 
But, on the other hand, it presupposes the generation of 
experience from physical changes. The world of sentient 
experience, according to the modern view, is closed and 
limited ; only the physical world is ample and large and 
of ever-to-be-discovered complexity. Plato's world of soul, 
on the other hand, is at least as large and ample as the 
world of things. 

Let us now try a crucial experiment. Can I form an 
intuition of a four-dimensional object ? Such an object 
is not given in the physical range of my sense contacts. 
All I can do is to present to myself the sequences of solids, 
which would mean the presentation to me under my con- 
ditions of a four-dimensional object. All I can do is to 
visualise and tactualise different series of solids which are 
alternative sets of sectional views of a four-dimensional 
shape. 

If now, on presenting these sequences, I find a power 
in me of intuitively passing from one of these sets of 
sequences to another, of, being given one, intuitively 
constructing another, not using a rule, but directly appre- 
hending it, then I have found a new fact about my soul, 
that it has a four-dimensional experience ; I have observed 
it by a function it has. 

I do not like to speak positively, for I might occasion 
a loss of time on the part of others, if, as may very well 



10 



A LANGirAGB OF SPACE 



A LANGUAGE OF SPACE 



11 



be, I am mistaken. But for my own i>art, I think there 
are indications of such an intuition ; from the results of 
my experiments, I adopt the hypothesis that that which 
thinks in us has an ample experience, of which the intui- 
tions we use in dealing with the world of real objecta 
are a part; of which experience, the intuition of four- 
dimensional forms and motions is also a part. The process 
we are engaged in intellectually is the reading the obscure 
signals of our nerves into a world of reality, by means of 
intuitions derived from the inner experience. 

The image I form is as follows. Imagine the captain 
of a modern battle-ship directing its course. He ha» 
his charts before him ; he is in communication with his 
associates and subordinates ; can convey his messages and 
commands to every part of the ship, and receive informa*- 
tion from the conning-tower and the engine-room. Now 
suppose the captain immersed in the problem of the 
navigation of his ship over the ocean, to have so absorbed 
himself in the problem of the direction of his craft over 
the plane surface of the sea that he forgets himself. All 
that occupies his attention is the kind of movement that 
his ship makes. The operations by which that movement 
is produced have sunk below the threshold of his con- 
sciousness, his own actions, by which he pushes the buttons, 
gives the orders, are so familiar as to l)e automatic, his 
mind is on the motion of the ship as a whole. In such 
a case we can imagine that he identifies himself with his 
ship ; all that enters his conscious thought is the direction 
of its movement over the plane surface of the ocean. 

Such is the relation, as I imagine it, of the soul to the 
body. A relation which we can imagine as existing 
momentarily in the case of the captain is the normal 
one in the case of the soul with its craft. As the captain 
is capable of a kind of movement, an amplitude of motion, 
which does not enter into his thoughts with regard to the 



^ > 



il. 



• It . 



directing the ship over the plane surface of the ocean, so 
the soul is capable of a kind of movement, has an ampli- 
tude of motion, which is not used in its task of directing 
the body in the three-dimensional region in which the 
body's activity lies. If for any reason it became necessary 
for the captain to consider three-dimensional motions with 
regard to his ship, it would not be difficult for him to 
gain the materials for thinking about such motions ; all 
he has to do is to call his own intimate experience into 
play. As far as the navigation of the ship, however, is 
concerned, he is not obliged to call on such experience. 
The ship as a whole simply moves on a surface. The 
problem of three-dimensional movement does not ordinarily 
concern its steering. And thus with regard to ourselves 
all those movements and activities which characterise our 
bodily organs are three-dimensional ; we never need to 
consider the ampler movements. But we do more than 
use the movements of our body to effect our aims by 
direct means ; we have now come to the pass when we act 
indirectly on nature, when we call processes into play 
which lie beyond the reach of any explanation we can 
give by the kind of thought which has been sufficient for 
the steering of our craft as a whole. When we come to 
the problem of what goes on in the minute, and apply 
ourselves to the mechanism of the minute, we find our 
habitual conceptions inadequate. 

The captain in us must wake up to his own intimate 
nature, realise those functions of movement which are his 
own, and in virtue of his knowledge of them apprehend 
how to deal with the problems he has come to. 

Think of the history of man. When has there been a 
time, in which his thoughts of form and movement were 
not exclusively of such varieties as were adapted for his 
bodily performance? We have never had a demand to 
conceive what our own most intimate powers are. But^ 



Ii 



12 



A LANGUAGE OF SPACE 



A LANGUAGE OF SPACE 



13 



just as little as by immersing himself in the steering of 
his ship over the plane surface of the ocean, a captain 
can loose the faculty of thinking about what he actually 
does, so little can the soul loose its own nature. It 
can be roused to an intuition that is not derived from 
the experience which the senses give. All that is 
necessary is to present some few of those appearances 
which, while inconsistent with three-dimensional matter, 
are yet consistent with our formal knowledge of four- 
dimensional matter, in order for the soul to wake up and 
not begin to learn, but of its own intimate feeling fill up 
the gaps in the presentiment, grasp the full orb of possi- 
bilities from the isolated points presented to it. In relation 
to this question of our perceptions, let me suggest another 
illustration, not taking it too seriously, only propounding 
it to exhibit the possibilities in a broad and general way. 

In the heavens, amongst the multitude of stars, there 
are some which, when the telescope is directed on them, 
seem not to be single stars, but to be split up into two. 
Regarding these twin stars through a spectroscope, an 
astronomer sees in each a spectrum of bands of colour and 
black lines. Comparing these spectrums with one another, 
he finds that there is a slight relative shifting of the dark 
lines, and from that shifting he knows that the stars are 
rotating round one another, and can tell their relative 
velocity with regard to the earth. By means of his 
terrestrial physics he reads this signal of the skies. This 
shifting of lines, the mere slight variation of a black line 
in a spectrum, is very unlike that which the astronomer 
knows it means. But it is probably much more like what 
it means than the signals which the nerves deliver are 
like the phenomena of the outer world. 

No picture of an object is conveyed through the nerves. 
No picture of motion, in the sense in which we postulate 
its existence, is conveyed through the nerves. The actual 






• 4,'^ 



. i>- 



tV 



deliverances of which our consciousness takes account are 
probably identical for eye and ear, sight and touch. 

If for a moment I take the whole earth together and 
regard it as a sentient being, I find that the problem of 
its apprehension is a very complex one, and involves a 
long series of personal and physical events. Similarly the 
problem of our apprehension is a very complex one. I 
only use this illustration to exhibit my meaning. It has 
this especial merit, that, as the process of conscious 
apprehension takes place in our case in the minute, so, 
with regard to this earth being, the corresponding process 
takes i)lace in what is relatively to it very minute. 

Now, Plato's view of a soul leads us to the hypothesis 
that that which we designate as an act of apprehension 
may be a very complex event, both physically and per- 
sonally. He does not seek to explain what an intuition 
is; he makes it a basis from whence he sets out on a 
voyage of discovery. Knowledge means knowledge ; he 
puts conscious being to account for conscious being. He 
makes an hypothesis of the kind that is so fertile in 
physical science — an hypothesis making no claim to 
finality, which marks out a vista of possible determination 
behind determination, like the hypothesis of space itself, 
the type of serviceable hypotheses. 

And, above all, Plato's hypothesis is conducive to ex- 
periment. He gives the perspective in which real objects 
can be determined ; and, in our present enquiry, we are 
making the simplest of all possible experiments — we are 
enquiring what it is natural to the soul to think of matter 
as extended. 

Aristotle says we always use a *' phantasm " in thinking, 
a phantasm of our corix)real senses a visualisation or a 
tactualisation. But we can so modify that visualisation 
or tactualisation that it represents something not known 
by the senses. Do we by that representation wake up an 



14 



A LANGUAGE OF SPACE 



A LANGUAGE OF SPACE 



15 



en 



cl 



intuition of the soul? Can we by the presentation of 
these hypothetical forms, that are the subject of our 
present discussion, wake ourselves up to higher intuitions ? 
And can we explain the world around by a motion that we- 
only know by our souls ? 

Apart from all speculation, however, it seems to me 
that the interest of these four-dimensional shapes and 
motions is sufficient reason for studying them, and that 
they are the way by which we can grow into a fuller 
apprehension of the world as a concrete whole. 

Space Names. 

If the words written in the squares drawn in fig. 1 are 
used as the names of the squares in the positions in 

which they are placed, it is evident that 
a combination of these names will denote 
a figure composed of the designated 
squares. It is found to be most con- 
venient to take as the initial square that 
marked with an asterisk, so that the 
Fig. 1. directions of progression are towards the 

observer and to his right. The directions 
of progression, however, are arbitrary, and can be chosen 
at will. 

Thus ety atj it^ an, al will denote a figure in the form 
of a cross composed of ^ve squares. 

Here, by means of the double sequence, efa^i and rtyt^ly it, 
is possible to name a limited collection of space elements. 
The system can obviously be extended by using letter 
sequences of more members. 

But, without introducing such a complexity, the 
principles of a space language can be exhibited, and a 
nomenclature obtained adequate to all the consideration* 
of the preceding pages. 



al 



> ^^^k.' 



*^t* 






1. Extension. 

Call the large squares in fig. 2 by the name written 

in them. It is evident that each 
can be divided as shown in fig. 1» 
Then the small square marked 1 
will be "en" in "En," or " Enen. 
The square marked 2 will be **et 
in " En " or '* Enet," while the 
square marked 4 will be "en " in 
" Et " or " Eten." Thus the square 
5 will be called " Ilil." 

This principle of extension can 



1 


2 


J 


ii 


El 




En 




Et 








An 


Al 


Al 


In 


h 


II 



if 



ir 



Fig. 2. 

be applied in any number of dimensions. 



Icn 


let 


kl 


Ian 


lat 


lal 


Ini 


lU 


r*i 



2. Application to Three-DiToefn^ional Space. 
To name a three-dimensional collocation of cubes take 
the upward direction first, secondly the 
direction towards the observer, thirdly the 
direction to his right hand. 

These form a word in which the first 
letter gives the place of the cube upwards^ 
the second letter its place towards the 
observer, the third letter its place to the 
right. 

We have thus the following scheme^ 
which represents the set of cubes of 
column 1, fig. 101, page 165. 

We begin with the remote lowest cube 
at the left hand, where the asterisk is 
placed (this proves to be by far the most 
convenient origin to take for the normal 
system). 

Thus **nen" is a "null" cube, ^Hen 
a red cube on it, and " len " a " null 



ten 


let 


tel 


tan 


tat 


tat 


'till 


til 


III 



« 


nen 


net 


nel 




nan 


nat 


nai 




nin 


nit 


nil 



iy 



y> 



cube above " ten." 



16 



A LANGUAGE OF SPACE 



Bj using a more extended sequence of consonants and 
vowels a larger set of cubes can be named. 

To name a four-dimensional block of tesserae ts it is 
simply necessary to prefix an " e," an ** a," or an ** i " to 
the cube names. 

Thus the tesseract blocks schematically represented on 
page 165, fig. 101 are named as follows : — 



elcn 


elci 


del 




ale 11 


alct 


a lei 




ilen 


ilet 


ilel' 


tlan 


clat 


clal 


a Ian 


alat 


alal 


ilan 


ilat 


ilal 


elm 


clU 


elll 


aim 


alit 


alii 


ilin 


ilit 


ilil 



eien 


ciet 


etel: 




aten 


atet 


atel 




iten 


itei 


itcl 


ftai» 


ctat 


etal 


a tan 


atat 


atal 


itan 


itat 


iul 


Ctiil 


clit 


elll 


atin 


alit 


atil 


itin 


itit 


itil 



* 


cuen 


enet 


enel 




cnan 


cnat 


eiial 




cnin 


cnit 


enil 



anen 


a net 


apel 


nnan 


anat 


anal 


anin 


anit 


anil 



inen 


inei. 


inel 


'man 


inat 


inal 


inin 


init 


Inil 



!2. Derivation of Point, Line, Face, etc., Names. 

The principle of derivation can be shown as follows 
Taking the square of squares 



en 


et 


rl 


an 


at 


al 


in 


it 


il 



A LANGUAGE OF SPACE 



17 



>. 



' 4' 



the number of squares in it can be enlarged and the 
whole kept the same size. 



1 

en 


ct 


et 


el 


an 


at 


at 


al 


an 


at 


at 


al 


in 


it 


it 


il 



Compare fig. 79, p. 138, for instance, or the bottom layer 

of fig. 84. 

Now use an initial '* s " to denote the result of carrying 
this process on to a great extent, and we obtain the limit 
names, that is the point, line, area names for a square, 
" Sat " is the whole interior. The corners are " sen,** 

"sel," "sin," " sil," while the lines 



5Cil 



Win 




are " san," "sal," "set," "sit." 



I find that by the use of the 
initial *' s " these names come to be 
practically entirely disconnected with 
the systematic names for the square 
from which they are derived. They 
are easy to learn, and when learned 
can be used readily with the axes running in any 
direction. 

To derive the limit names for a four-dimensional rect- 
angular figure, like the tesseract, is a simple extension of 
this process. These point, line, etc., names include those 
which apply to a cube, as will be evident on inspection 
of the first cube of the diagrams which follow. 

All that is necessary is to place an '* s " before each of the 
names given for a tesseract block. We then obtain 
apellatives which, like the colour names on page 174, 
fig. 103, apply to all the points, lines, faces, solids, and to 



18 



A LANGUAGE OF SPACE 



A LANGUAGE OF SPACE 



19 



the hypersolid of the tesseract. These names have the 
advantage over the colour marks that each point, line, etc., 
has its own individual name. 

In the diagrams I give the names corresponding to 
the positions shown in the coloured plate or described on 
p. 174. By comparing cubes 1, 2, 3 with the first row of 
cubes in the coloured plate, the systematic names of each 
of the points, lines, faces, etc., can be determined. The 
asterisk shows the origin from which the names run. 

These point, line, face, etc., names should be used in 
•connection with the corresponding colours. The names 
should call up coloured images of the parts named in their 
right connection. 

It is found that a certain abbreviation adds vividness of 
•distinction to these names. If the final " en " be dropped 
wherever it occurs the system is improved. Thus instead 
of " senen," "seten," " selen," it is preferable to abbreviate 
to " sen," " set," " sel," and also use '' san," ** sin " for 



*' sanen," " sinen." 



3ele n S^Ut Stld S.U SAt\ S.M Sihn Sdtl S.U 

2 0. x<:. Stiit 



^7" 






^^ 



M 




•V « 



5mi^ 



Saui} 



Sinit 









C« 


Sihl 






<» 




«— < 


Sintl S. 


mil 



V 



ian^t 






m Stner iS'Htl 



% 



Intdtar Jef.'t 



Inhtrior Jatjf 



5:ui^ x*/ 



iiihtrior J I ^a t 



r 



r . 



[ 



ill __Siia_iilel 5a an blUX 5llal 5clm 5\\\\ Sli.'l 



Co 



•J!n 



^^ Jaltt 




rt 0«iun 



r^ 






v. 



^^. 



St-^.' 







orntt 



Sena 




btmi 



StniX 



St nil 



bxxtX S 












5an«.t 



h\m S 



Omit Ji'iu'l 



\ 



'/ 



5.imX ^ 



Jnlcm> Odttt 




lurfinpr ^aldT 



JnUrior 0^^^ 






in 







6dnlT 



oamT" 



vic-rul 




i-ra r 



iantt" A»n«t 






tSaTut 



< 



V 



inferior SaXi 






3aTl.J 5»n>] 



'^<- 

V 



rb 



inhrldr OJ^a^ 



tSarul 



Jnlirior Oi^al 




20 



A LANGUAGE OF SPACE 



/ 



A LANGUAGE OF SPACE 



21 



/• 



// 



tx 



<ScTt Ofntt xStnfct 



Co 



-% 3enat\^ 



St 



^Jtnin S«iiiK — 













T^TT 






5tl 



•«-' 






% 



isita Niii 



t/i 



Fi 54ltt r 



im. 



iiUT 



5a.! 



6anet ; 
< 

5inet 3lnet 



'^ 



V 



jinat 



%. 













5iUt 




inTcTioT oanAl 



inherior 6a tat 



jntirior Jitat 



We can now name any section. Take e.g. the line in 
the first cube from senin to senel, we should call the line 
running from senin to senel, senin senat senel, a line 
light yellow in colour with null points. 

Here senat is the name for all of the line except its ends. 
Using " senat " in this way does not mean that the line is 
the whole of senat, but what there is of it is senat. It is 
a part of the senat region. Thus also the triangle, which 
has its three yertices in senin, senel, selen,is named thus : 

Area : setat. 

Sides : setan, senat, setet. 

Vertices : senin, senel, sel. 

The tetrahedron section of the tesseract can be thought 
of as a series of plane sections in the successiye sections of 
the tesseract shown in fig. 114, p. 191. In bo the section 
is the one written above. In \ the section is made by a 



plane which cuts the three edges from sanen intermediate 
of their lengths and thus will be : 

Area : satat. 

Sides : satan, sanat, satet. 

Vertices : sanan, sanet, sat. 

The sections in \\^ bg will be like the section in bj but 
smaller. 

Finally in b4 the section plane simply passes through the 
corner named sin. 

Hence, putting these sections together in their right 
relation, from the face setat, surrounded by the lines and 
points mentioned above, there run : 

3 faces: satan, sanat, satet 
3 lines : sanan, sanet, sat 

and these faces and lines run to the point sin. Thus 
the tetrahedron is completely named. 

The octahedron section of the tesseract, which can be 
traced from fig. 72, p. 129 by extending the lines there 
drawn, is named : 

Front triangle selin, selat, selel, setal, senil, setit, selin 
with area setat. 

The sections between the front and rear triangle, of 
which one is shown in lb another in 2b, are thus named, 
points and lines, salan, salat, salet, satet, satel, satal, sanal, 
sanat, sanit, satit, satin, satan, salan. 

The rear triangle found in 3b by producing lines is sil, 
sitet, sinel, sinat, sinin, sitan, sil. 

The assemblage of sections constitute the solid body of 
the octahedron satat with triangular faces. The one from 
the line selat to the point sil, for instance, is named 



22 



»»< 



A LANGUAGE OF SPACE 



A LANGUAGE OF SPACE 



23 



selin, selat, selel, salet, salat, salan, sil. The whole 
interior is salat. 

Shapes can easily be cut out of cardboard which, when 
folded together, form not only the tetrahedron and the 
octohedron, but also samples of all the sections of the 
tesseract taken as it i^asses comerwise through our sjiace. 
To njime and visualise with appropriate colours a series of 
these sections is an admirable exercise for obtainincr 
familiarity with the subject. "^ 



Extension and Connection with Numbers. 

By extending the letter sequence it is of course possible 
to name a larger field. By using the limit names the 
corners of each square can be named. 

Thus " en sen," - an sen," etc., will be the names of the 
points nearest the origin in " en " and in ** an." 

A field of points of which each one is indefinitely small 
IS given by the names written below. 



*. 


? f , 


[ensen 


eiscn iciscn; 


f < 


» 4 i 


laiiscn 


aiscn laUen j 


------H 


1 • 
1 • 


• insen 


itscn : ilseii I 
1 1 



The squares are shown in dotted lines, the names 
denote the points. These points are not mathematical 
points, but really minute areas. 

Instead of starting with a set of squares and naming 
them, we can start with a set of points. 

By an easily remembered convention we can give 
names to such a region of points. 



Let the space names with a final " e " added denote the 
mathematical points at the corner of each square nearest 
the origin. We have then 



O 1 2 

or •----r-— 1 

iene 'Cte ;eie 
I 



lane ate Sale 

: • ! 



at.— -^.— --»-.-,-- 
'ine ;ite ; ile 

i I • 



.i J 



for the set of mathematical points indicated. This 
system is really completely independent of the area 
system and is connected with it merely for the purpose 
of facilitating the memory processes. The word " ene " is 
pronounced like "eny," with just sufficient attention to 
the final vowel to distinguish it from the word " en." 

Now, connecting the numbers 0, 1, 2 with the sequence 
e, a, i, and also with the sequence n, t, I, we have a set of 
points named as with numbers in a co-ordinate system. 
Thus *'ene" is (0, 0) '^ate" is (1, 1) ^'ite" is (2, 1). 
To pass to the area system the rule is that the name of 
the square is formed from the name of its point nearest 
to the origin by dropping the final e. 

By using a notation analogous to the decimal system 
a larger field of points can be named. It remains to 
assign a letter sequence to the numbers from positive 
to positive 9, and from negative to negative 9, to obtain 
a system which can be used to denote both the usual 
co-ordinate system of mapping and a system of named 
squares. The names denoting the points all end with e. 
Those that denote squares end with a consonant. 

There are many considerations which must be attended 
to in extending the sequences to be used, such as 
uniqueness in the meaning of the words formed, ease 
of pronunciation, avoidance of awkward combinations. 



24 



A LANGUAGE OF SPACE 



I drop 



"s" 



short **u" 



altogether from the consonant series and 
from the vowel series. It is convenient to 
have unsignificant letters at disposal. A double consonant 
like " st " for instance can be referred to without giving it 
a local significance by calling it "ust." I increase the 
number of vowels by considering a sound like " ra " to 
be a vowel, using, that is, the letter "r" as forming a 
compound vowel. 

The series is as follows : — 





# 




Consonants. 










' 


positive 
negative 




n 
z 


1 

t 
d 


2 

1 
th 


3 4 5 
p f sh 
b v m 

Vovy:l.LS. 


6 
k 

g 


7 
ch 

• 


8 

nt 

nd 


9 
st 
sp 


\ 


positive 
negative 



e 
er 


1 
a 
o 


2 

• 

1 
oo 


3 4 5 
ee ae ai 
io oe iu 


6 
ar 
or 


7 
ra 
ro 


8 

• 

n 
roo 


9 

ree 
rio 


^ 



e as in men ; a as m man ; i as m m ; 



\ 



Trcmunciaiiim. 
ee as in between ; ae as ay in may ; ai as i in mine ; ar as 
in art ; er as ear in earth ; o as in on ; oo as oo in soon ; 
io as in clarion ; oe as oa in oat ; iu pronounced like yew. 

To name a point such as (23, 41) it is considered as 
(3, 1) on from (20, 40) and is called "- ifeete." It is the 
initial point of the square ifeet of the area system. 

The preceding amplification of a space language has 
been introduced merely for the sake of completeness. As 
has already been said nine words and their combinations, 
applied to a few simple models suffice for the purposes of 
our present enquiry. 



London: SWAN SONNENSCHEIN & CO., LIMITED, 
25 High Street, Bloomsbury. 



Printed by HaMeU, fVatsoM &• Vinty, Ld,* 4-*^" ""^ Aylesbury, 



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